MATRIX THEORY CHAPTER 3 FALL 2017 1.JORDAN FORM 1.1.Main Theorem 「A1 入1 Jm(A)= Here JA is called the Jordan form of A,which is unique up to a permutation of the blocks Here are several basic questions (a)What the Jordan form JA can tell us about the original matrix A. (b)Given A,how to find its Jordan form A? to nn Example 1.Suppose A has a Jordan form 「21 J=2(2)⊕2(3)⊕2(3)⊕J(3) 31 3 31 1 31 3 (a)Characte The algeb =2,2=3 (b)For A1 =2 rank(A-2）=rank(J-2）=9, rank(A-21)2 rank(J-2)2=8, rank(A-21)=rank(J)=8,k MATRIX THEORY - CHAPTER 3 FALL 2017 1. Jordan form 1.1. Main Theorem. Theorem 1. Given any matrix A ∈ Mn, there exists a nonsingular matrix S such that S −1AS = JA, where JA is a block diagonal matrix with each block in the form Jm(λ) =        λ 1 λ 1 . . . . . . λ 1 λ        Here JA is called the Jordan form of A, which is unique up to a permutation of the blocks. Here are several basic questions: (a) What the Jordan form JA can tell us about the original matrix A. (b) Given A, how to find its Jordan form A? (c) Given A, how to find the nonsingular matrix S such that AS = SJA. (d) Where the Jordan form can be used? Example 1. Suppose A has a Jordan form J = J2(2) ⊕ J2(3) ⊕ J2(3) ⊕ J4(3) =                 2 1 2 3 1 3 3 1 3 3 1 3 1 3 1 3                 (a) Characteristic polynomial: pA(t) = pJ (t) = (t−2)2 (t−3)8 . There are two eigenvalues λ1 = 2, λ2 = 3. The algebraic multiplicity for λ1 = 2 is 2 and the algebraic multiplicity for λ2 = 3 is 8. (b) For λ1 = 2 rank(A − 2I) = rank(J − 2I) = 9, rank(A − 2I) 2 = rank(J − 2I) 2 = 8, rank(A − 2I) k = rank(J − 2I) k = 8, ∀k ≥ 2 2 is the largest block dimension with diagonal λ1 = 2. 10 − 9 = 1 is the number of blocks with diagonal λ1 = 2 1